Integrand size = 23, antiderivative size = 23 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Int}\left (\frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 394, normalized size of antiderivative = 17.13 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {-i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+12 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]+\frac {24 \cos (c+d x) (a+b \sin (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}}{18 a b d} \]
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Time = 1.74 (sec) , antiderivative size = 241, normalized size of antiderivative = 10.48
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}+\frac {2}{3 b}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} b +\textit {\_R}^{3} a +\textit {\_R} a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 a b}}{d}\) | \(241\) |
default | \(\frac {\frac {-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}+\frac {2}{3 b}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} b +\textit {\_R}^{3} a +\textit {\_R} a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 a b}}{d}\) | \(241\) |
risch | \(-\frac {2 \,{\mathrm e}^{i \left (d x +c \right )} \left (2 i a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 i a \,{\mathrm e}^{i \left (d x +c \right )}-b \right )}{3 a b d \left (b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (531441 a^{10} b^{8} d^{6} \textit {\_Z}^{6}+59049 a^{8} b^{6} d^{4} \textit {\_Z}^{4}+2187 a^{6} b^{4} d^{2} \textit {\_Z}^{2}+a^{6}+15 a^{4} b^{2}+48 a^{2} b^{4}-64 b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {118098 a^{9} b^{7} d^{5} \textit {\_R}^{5}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\left (\frac {6561 i d^{4} b^{5} a^{9}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\frac {52488 i d^{4} b^{7} a^{7}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right ) \textit {\_R}^{4}+\left (\frac {11664 a^{7} b^{5} d^{3}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}-\frac {11664 d^{3} b^{7} a^{5}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right ) \textit {\_R}^{3}+\left (\frac {486 i d^{2} b^{3} a^{7}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\frac {3888 i d^{2} b^{5} a^{5}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right ) \textit {\_R}^{2}+\left (-\frac {9 d b \,a^{7}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\frac {342 a^{5} b^{3} d}{a^{6}-48 a^{2} b^{4}+128 b^{6}}-\frac {576 d \,b^{5} a^{3}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right ) \textit {\_R} +\frac {9 i a^{5} b}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\frac {72 i b^{3} a^{3}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right )\right )\) | \(584\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 9.40 (sec) , antiderivative size = 9984, normalized size of antiderivative = 434.09 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
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Not integrable
Time = 148.60 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )}}{\left (a + b \sin ^{3}{\left (c + d x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 6.11 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {\cos \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \]
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Time = 15.05 (sec) , antiderivative size = 2431, normalized size of antiderivative = 105.70 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
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